"I've got a lot of family here tonight, and when I say that, I don't just mean kids, I mean grandkids!" says Paul McCartney, perennially trim and dapper at 69. He scratches his suspiciously dark hair in mock-wonderment. "What must they be thinking? 'There's grandad up there, rockin' away?'"

Yet the lucky McCartney progeny will not be troubled by the problem of perception that for years saw Grandad derided as the un-hip former Beatle, the populist square, lacking John Lennon's iconoclastic cool. Like the rest of the arena, they will have enjoyed a consummate live show that was near perfection.

Heavy on Beatles classics and light on new material, tonight is essentially the live show McCartney has been touring on and off for the last decade. But that familiarity does not breed contempt for his peerless back catalogue. It helps that he tackles timeless works of melodic magnificence such as Paperback Writer and Eleanor Rigby with the same fervour that he did 40 years ago.

His goofing and between-song patter is likewise so slick that it can border on shtick, yet still comes across as sincere. McCartney's eulogy to Lennon before Here Today, regretting their fallout, remains a tearjerker, while an equally moving tribute to George Harrison sees him strum Something on a ukulele.

It's lucky McCartney has charm to burn, as his eccentric stage antics – vigorous bouts of self-mocking, dad-dancing, and a cod-Rasta accent to introduce his side project, the Fireman's Sing the Changes – would otherwise have his grandkids, and indeed the whole arena, cringing. Genius does not normally come in such genial packages, but genius it undoubtedly is: The Long and Winding Road remains an exquisite meditation on mortality, and you could hear a pin drop during a breathtaking a cappella Blackbird, dedicated to the 1960s US civil rights movement.

A late-set sequence of songs including All You Need Is Love, Day Tripper, Get Back – featuring a cadaverous Ronnie Wood – and Yesterday is so effortlessly magnificent, it borders on showboating. Pop gets no more rarefied, although one middle-aged fan filing out of the arena is still not satisfied: "He was great," she says to her companion. "But fancy not playing Imagine."